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The integration of physics-based knowledge with machine learning models is increasingly shaping the monitoring, diagnostics, and prognostics of electrical transformers. In this two-part series, the first paper introduced the foundations of…

Machine Learning · Computer Science 2025-12-30 Jose I. Aizpurua

We present pseudo-differential enhanced physics-informed neural networks (PINNs), an extension of gradient enhancement but in Fourier space. Gradient enhancement of PINNs dictates that the PDE residual is taken to a higher differential…

Machine Learning · Computer Science 2026-05-06 Andrew Gracyk

In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of…

Numerical Analysis · Mathematics 2024-09-06 Jihahm Yoo , Haesung Lee

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integral equations with sparse, noisy, unstructured, and multi-fidelity data. PINNs incorporate all available information into a…

Analysis of PDEs · Mathematics 2020-04-10 Guofei Pang , Marta D'Elia , Michael Parks , George E. Karniadakis

This paper explores the possibilities of applying physics-informed neural networks (PINNs) in topology optimization (TO) by introducing a fully self-supervised TO framework that is based on PINNs. This framework solves the forward…

Computational Engineering, Finance, and Science · Computer Science 2022-12-19 Junyan He , Shashank Kushwaha , Charul Chadha , Seid Koric , Diab Abueidda , Iwona Jasiuk

The solution of partial differential equations (PDES) on irregular domains has long been a subject of significant research interest. In this work, we present an approach utilizing physics-informed neural networks (PINNs) to achieve…

Computational Physics · Physics 2025-06-12 Cuizhi Zhou , Kaien Zhu

Physics Informed Neural Networks (PINNs) have recently gained popularity for solving partial differential equations, given the fact they escape the curse of dimensionality. In this paper, we present Physics Informed Neural Networks as an…

Numerical Analysis · Mathematics 2022-07-26 Umberto Zerbinati

Parameter estimation remains a challenging task across many areas of engineering. Because data acquisition can often be costly, limited, or prone to inaccuracies (noise, uncertainty) it is crucial to identify sensor configurations that…

Machine Learning · Statistics 2025-11-20 Georgios Venianakis , Constantinos Theodoropoulos , Michail Kavousanakis

In this paper we employ the emerging paradigm of physics-informed neural networks (PINNs) for the solution of representative inverse scattering problems in photonic metamaterials and nano-optics technologies. In particular, we successfully…

Computational Physics · Physics 2020-04-22 Yuyao Chen , Lu Lu , George Em Karniadakis , Luca Dal Negro

Maxwell's equations are a collection of coupled partial differential equations (PDEs) that, together with the Lorentz force law, constitute the basis of classical electromagnetism and electric circuits. Effectively solving Maxwell's…

Machine Learning · Computer Science 2023-08-15 Shiyuan Piao , Hong Gu , Aina Wang , Pan Qin

Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g.,…

Machine Learning · Computer Science 2024-10-22 Hamid El Bahja , Jan Christian Hauffen , Peter Jung , Bubacarr Bah , Issa Karambal

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods…

Numerical Analysis · Mathematics 2022-02-02 N. Sukumar , Ankit Srivastava

Physics-Informed Neural Networks (PINNs) have gained significant attention for their simplicity and flexibility in engineering and scientific computing. In this study, we introduce a normalized PINN (NPINN) framework to solve a class of…

Numerical Analysis · Mathematics 2025-03-11 Jichao Ma , Dandan Liu , Jinran Wu , Xi'an Li

We use Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a…

A physics-informed neural network (PINN) is developed, for the first time, to learn the time-dependent quasi-static magnetohydrodynamic (MHD) equations in axisymmetric tokamak geometry, without any experimental or synthetic data. The…

Plasma Physics · Physics 2026-04-23 Jonathan S. Arnaud , Christopher J. McDevitt , Golo Wimmer , Xian-Zhu Tang

Temporally and spatially dependent uncertain parameters are regularly encountered in engineering applications. Commonly these uncertainties are accounted for using random fields and processes, which require knowledge about the appearing…

Numerical Analysis · Mathematics 2021-11-23 Jan Niklas Fuhg , Ioannis Kalogeris , Amélie Fau , Nikolaos Bouklas

This paper introduces a framework based on physics-informed neural networks (PINNs) for addressing key challenges in nonlinear lattices, including solution approximation, bifurcation diagram construction, and linear stability analysis. We…

Numerical Analysis · Mathematics 2025-07-22 Muhammad Luthfi Shahab , Fidya Almira Suheri , Rudy Kusdiantara , Hadi Susanto

Interest is rising in Physics-Informed Neural Networks (PINNs) as a mesh-free alternative to traditional numerical solvers for partial differential equations (PDEs). However, PINNs often struggle to learn high-frequency and multi-scale…

Machine Learning · Computer Science 2025-02-25 Madison Cooley , Varun Shankar , Robert M. Kirby , Shandian Zhe

Physics-Informed Neural Networks (PINNs) have received increased interest for forward, inverse, and surrogate modeling of problems described by partial differential equations (PDE). However, their application to multiphysics problem,…

Machine Learning · Computer Science 2022-09-09 Danial Amini , Ehsan Haghighat , Ruben Juanes

Physics-informed neural networks (PINNs) commonly address ill-posed inverse problems by uncovering unknown physics. This study presents a novel unsupervised learning framework that identifies spatial subdomains with specific governing…

Machine Learning · Computer Science 2024-12-11 Arturo Rodriguez , Ashesh Chattopadhyay , Piyush Kumar , Luis F. Rodriguez , Vinod Kumar